Related papers: Euler's factorial series and global relations

Any power series with unit constant term can be factored into an infinite product of the form $\prod_{n\geq 1} (1-q^n)^{-a_n}$. We give direct formulas for the exponents $a_n$ in terms of the coefficients of the power series, and vice…

When $p$ is an odd prime, Delbourgo observed that any Kubota-Leopoldt $p$-adic $L$-function, when multiplied by an auxiliary Euler factor, can be written as an infinite sum. We shall establish such expressions without restriction on $p$,…

Let $p>3$ be a prime, and let $a$ be a rational p-adic integer with $a\not\equiv 0\pmod p$. In this paper we establish congruences for $$\sum_{k=1}^{(p-1)/2}\frac{\binom ak\binom{-1-a}k}k, \quad\sum_{k=0}^{(p-1)/2}k\binom ak\binom{-1-a}k…

In this paper, we estimate the simultaneous approximation exponents of the values of certain Mahler functions. For this we construct Hermite-Pad\'{e} approximations of the functions under consideration, then apply the functional equations…

In this note we associate a sequence of non-negative integers to any convergent series of positive real numbers and study this sequence for the series $\sum_{n \geq 1} n^{-k}$ where $k$ is an integer $\geq 2$.

We derive analytic solutions for the potential and field in a one-dimensional system of masses or charges with periodic boundary conditions, in other words Ewald sums for one dimension. We also provide a set of tools for exploring the…

In this paper we generalize and put in a new light part of ``Fouier analysis on Number fields and Hecke's zeta function''[14] by Tate. We express the relative Euler characteristic using purely adelic language. By using certain natural…

We introduce a Dirichlet-series framework for studying the asymptotic behavior of generalized factorial functions defined by Legendre-type valuation formulas. Let $K$ be a number field and let $S$ be a finite set of prime ideals. For a…

We study the complexity of the classic capacitated k-median and k-means problems parameterized by the number of centers, k. These problems are notoriously difficult since the best known approximation bound for high dimensional Euclidean…

Inspired by the results of Rhin and Viola (2001), the purpose of this work is to elaborate on a series representation for $\zeta \left( 3\right)$ which only depends on one single integer parameter. This is accomplished by deducing a…

In this paper we introduce confluence relations for motivic Euler sums (also called alternating multiple zeta values) and show that all linear relations among motivic Euler sums are exhausted by the confluence relations. This determines all…

We investigate the rational approximation of fractional powers of unbounded positive operators attainable with a specific integral representation of the operator function. We provide accurate error bounds by exploiting classical results in…

We show that Hermite's approximations to values of the exponential function at given algebraic numbers are nearly optimal when considered from an adelic perspective. We achieve this by taking into account the ratio of these values whenever…

The partial fraction expansion of coth($\pi$z), due to Euler, is generalized to power series having for coefficients the Riemann zeta function evaluated at certain arithmetic sequences. A further generalization using arbitrary Dirichlet…

In this paper we introduce and study a family $\Phi_k$ of arithmetic functions generalizing Euler's totient function. These functions are given by the number of solutions to the equation $\gcd(x_1^2+\ldots +x_k^2, n)=1$ with $x_1,\ldots,x_k…

If $E$ is a subset of the integers then the $n$-th characteristic ideal of $E$ is the fractional ideal of $\mathbb{Z} $ consisting of $0$ and the leading coefficients of the polynomials in $\mathbb{Q}[x]$ of degree no more than $n$ which…

In the Euclidean $k$-Means problem we are given a collection of $n$ points $D$ in an Euclidean space and a positive integer $k$. Our goal is to identify a collection of $k$ points in the same space (centers) so as to minimize the sum of the…

Asymptotic expansions of series $\sum_{k=0}^\infty \epsilon^k(k+a)^\gamma e^{-(k+a)^\alpha x}$ and $\sum_{k=0}^\infty \epsilon^k(k+a)^\gamma / (x(k+a)^\alpha+1)^\mu}$ in powers of $x$ as $x\to+0$ are found, where $\epsilon=1$ or…

Let $f$ be a newform of even weight $2\kappa$ for $D^\times$, where $D$ is a possibly split indefinite quaternion algebra over $\mathbb{Q}$. Let $K$ be a quadratic imaginary field splitting $D$ and $p$ an odd prime split in $K$. We extend…

This article studies the singular values of entire functions of the form $E^k (z)+P(z)$ where $E^k$ denotes the $k-$times composition of $e^z$ with itself and $P$ is any non-constant polynomial. It is proved that the full preimage of each…